# algebra precalculus – Solutions for \$ 2 ^ x + 2 ^ y = 2 ^ z \$ for integers x, y and z

I'm trying to prove / disprove that the equation

$$2 ^ x + 2 ^ y = 2 ^ z$$

or $$x$$, $$y$$ and $$z$$ are positive integers, only have trivial solutions. The obvious case where this is true would be for $$x = y$$ but I'm not sure if solutions exist for $$x only$$

I have the impression that there are no other solutions, but I do not know how one could officially prove it. Someone has advice on how to do this?

Sorry it's about a trivial issue / a known problem.